Finding a number of twin primes less than a certain number I was doing some problems on number theory, and I came across the following question: "How many twin primes less than 100 exist?" I was wondering if anyone could tell me what method would be used to solve this problem and others too.
Thanks!
 A: *

*You can count the twin primes less than $n$ by hand.

*You can find the number of twin primes less than $p_k!!$ (the product of primes not exceeding the $k$th prime) using the sieve S as modified in S' in this question, but there may not be any twin primes beyond some very large $n$. In any event the size of the numbers makes computation (of the prime pairs, if not their cardinality) difficult for even modest $k.$

*For small $n$ the sieve S above is useful. For the original question, by S there are 3 pairs (n,n+2) co-prime to $2,3,5$ on an interval of length 30. Then we have 12 co-prime pairs on (1,120) and so on $[1,100]$  we cannot have more than 12 pairs of twin primes. We can quickly compute co-prime pairs and  we see that 10 pairs are in fact twin primes, 8 less than 100.

*Version II of Chen's 1973 theorem says that for any positive even integer $h$ there are infinitely many primes $p$ such that $p+h$ is either a prime or semiprime. Since $h$ can be 2 there are infinitely many such pairs which are potentially twin primes. Up to boundaries which would not be much use for the OP, this may be the best we can currently say regarding existence of infinitely many twin primes. This is just a gloss on 2 above.     
Halberstam and Richert's Sieve Methods is a really good resource for questions in this area.  
A: HINT.Only I think to see Wilson's theorem and (because of primes less that $100$) looking for solutions of the two congruences
$$(p-1)!+1\equiv 0 \pmod p$$ $$(p+1)!+1\equiv 0 \pmod {p+2}$$
Paying some attention this could work, I guess. However for primes $p$ larger, verification in tables of primes or twin primes could be the only "method"
